1. January 23, 2001Physics 8411 Elastic Scattering of Electrons by Nuclei We want to consider the elastic scattering of electrons by nuclei to see (i) how finite size can be accounted for, and (ii) how the electromagnetic potential (1/r) is converted to a quantum mechanical matrix element. The presentation here follows that of Perkins, Introduction of High Energy Physics, Third Edition, chapter 6. We will start by considering electrons and nuclei as spinless isolated particles with the nucleus at rest. We will later consider the nucleus as part of an atom, and we will review the effects of spin, etc.

2. January 23, 2001Physics 8412 Elastic Scattering of Spinless Electrons by Nuclei - 1 In first order perturbation theory, the transition rate is where Recall, with That is, and are plane waves.

3. January 23, 2001Physics 8413 Elastic Scattering of Spinless Electrons by Nuclei - 2 Begin the calculation formally: where Given a charge density then

4. January 23, 2001Physics 8414 Elastic Scattering of Spinless Electrons by Nuclei - 3 Just a bit of algebra gives: where we have defined the elastic scattering form factor

5. January 23, 2001Physics 8415 Elastic Scattering of Spinless Electrons by Nuclei - 4 With, and  the polar angle between and

6. January 23, 2001Physics 8416 Elastic Scattering of Spinless Electrons by Nuclei - Add Atomic Screening The term which depends on the nature of the potential: diverges as so we say that the range of the electromagnetic potential is infinite. Atoms have clouds of electrons as well as nuclei, and the effect is to modify the electric potential: and the exponential factor is referred to as screening.

7. January 23, 2001Physics 8417 Elastic Scattering of Spinless Electrons by Neutral Atoms -1 With the screening potential, the matrix element becomes: The charge density is significantly greater than zero only for. Therefore. recall that.

8. January 23, 2001Physics 8418 Elastic Scattering of Spinless Electrons by Neutral Atoms -2

9. January 23, 2001Physics 8419 Elastic Scattering of Spinless Electrons by Neutral Atoms -2

10. January 23, 2001Physics 84110 Elastic Scattering of Spinless Electrons by Neutral Atoms -3

11. January 23, 2001Physics 84111 For, When is ? Elastic Scattering of Spinless Electrons by Neutral Atoms -4

12. January 23, 2001Physics 84112 Form Factors and Central Potentials We can use the same formalism for and central potential that we used for the 1/r potential, and the form factor will emerge again as a factorizable term.

13. January 23, 2001Physics 84113 The Finite Rutherford Proton - 1 When scattering spinless electrons from protons, we found that for Using the notation electron + nucleus electron + nucleus With the assumptions that (i) the recoiling nucleus is non- relativistic, and (ii) that the nuclear recoil momentum is small compared to the momentum of the incident electron: at rest

14. January 23, 2001Physics 84114 The Finite Rutherford Proton - 2 One can approximate so that With,

15. January 23, 2001Physics 84115 Point-like Elastic Scattering - 1 Rutherford Scattering - non-relativistic quantum mechanics, first Born approximation, no spin or magnetic moments. Mott Scattering - spin 1/2 electrons, spin 0 protons, single photon exchange Dirac Scattering - spin 1/2 electrons, spin 1/2 protons, point-like, single photon exchange.

16. January 23, 2001Physics 84116 Point-like Elastic Scattering - 2 Rutherford Scattering: Mott Scattering: Dirac Scattering:

17. January 23, 2001Physics 84117 Rosenbluth Scattering -1 Rosenbluth extends the Dirac formula to a finite proton: where and

18. January 23, 2001Physics 84118 Rosenbluth Scattering -2 To extract the form factors, one may fix and vary the scattering angle.

19. January 23, 2001Physics 84119 Rosenbluth Scattering -3 Experimentally, the form factors obey a simple scaling law: